Parameterize [tex]C_R[/tex] by
[tex]\mathbf r(t)=(R\cos t,R\sin t)[/tex]
with [tex]0\le t\le2\pi[/tex].
Plug [tex]\mathbf r[/tex] into [tex]\mathbf G[/tex] and compute the differential [tex]\mathrm d\mathbf r[/tex]:
[tex]\mathbf G(\mathbf r(t))=\dfrac{(R\cos t,R\sin t)}{\sqrt{R^2-1}}[/tex]
[tex]\mathrm d\mathbf r=(-R\sin t,R\cos t)\,\mathrm dt[/tex]
Plug everything into the integral and evaluate:
[tex]\displaystyle\int_{C_R}\mathbf G\cdot\mathrm d\mathbf r=\int_0^{2\pi}0\,\mathrm dt=0[/tex]
